4 problems

Let $T$ be a first-order theory, and consider its countable models up to isomorphism. Vaught's conjecture. The number of countable isomorphism types of models of $T$ is either coun…

Let $T$ be an Ehrenfeucht theory, meaning a complete theory with more than one but only finitely many countable models. For a complete type over $\emptyset$, its multiplicity is th…

Let $T$ be a complete theory. A model $M$ is algebraic if every element of $M$ is algebraic over $\emptyset$. The algebraic-model conjecture. If $T$ has an algebraic model, then…

Let $T$ be a complete theory and suppose that a minimal model of $T$ is a model with no proper elementary substructure. Pillay's conjecture. If $T$ has a minimal model, then $T$ ha…